Precalculus Examples

$3 | x + 2 | = 2 | x - 1 | - 1$

Step 1

Rewrite the equation as $2 | x - 1 | - 1 = 3 | x + 2 |$ .

$2 | x - 1 | - 1 = 3 | x + 2 |$

Step 2

Add $1$ to both sides of the equation.

$2 | x - 1 | = 3 | x + 2 | + 1$

Step 3

Divide each term in $2 | x - 1 | = 3 | x + 2 | + 1$ by $2$ and simplify.

Tap for more steps...

Step 3.1

Divide each term in $2 | x - 1 | = 3 | x + 2 | + 1$ by $2$ .

$\frac{2 | x - 1 |}{2} = \frac{3 | x + 2 |}{2} + \frac{1}{2}$

Step 3.2

Simplify the left side.

Tap for more steps...

Step 3.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.2.1.1

Cancel the common factor.

$\frac{2 | x - 1 |}{2} = \frac{3 | x + 2 |}{2} + \frac{1}{2}$

Step 3.2.1.2

Divide $| x - 1 |$ by $1$ .

$| x - 1 | = \frac{3 | x + 2 |}{2} + \frac{1}{2}$

Step 4

Solve for $| x + 2 |$ .

Tap for more steps...

Step 4.1

Rewrite the equation as $\frac{3 | x + 2 |}{2} + \frac{1}{2} = | x - 1 |$ .

$\frac{3 | x + 2 |}{2} + \frac{1}{2} = | x - 1 |$

Step 4.2

Subtract $\frac{1}{2}$ from both sides of the equation.

$\frac{3 | x + 2 |}{2} = | x - 1 | - \frac{1}{2}$

Step 4.3

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 | x + 2 |}{2} = \frac{2}{3} (| x - 1 | - \frac{1}{2})$

Step 4.4

Simplify both sides of the equation.

Tap for more steps...

Step 4.4.1

Simplify the left side.

Tap for more steps...

Step 4.4.1.1

Simplify $\frac{2}{3} \cdot \frac{3 | x + 2 |}{2}$ .

Tap for more steps...

Step 4.4.1.1.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.1.1.1.1

Cancel the common factor.

$\frac{2}{3} \cdot \frac{3 | x + 2 |}{2} = \frac{2}{3} (| x - 1 | - \frac{1}{2})$

Step 4.4.1.1.1.2

Rewrite the expression.

$\frac{1}{3} (3 | x + 2 |) = \frac{2}{3} (| x - 1 | - \frac{1}{2})$

Step 4.4.1.1.2

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.4.1.1.2.1

Factor $3$ out of $3 | x + 2 |$ .

$\frac{1}{3} (3 | x + 2 |) = \frac{2}{3} (| x - 1 | - \frac{1}{2})$

Step 4.4.1.1.2.2

Cancel the common factor.

$\frac{1}{3} (3 | x + 2 |) = \frac{2}{3} (| x - 1 | - \frac{1}{2})$

Step 4.4.1.1.2.3

Rewrite the expression.

$| x + 2 | = \frac{2}{3} (| x - 1 | - \frac{1}{2})$

Step 4.4.2

Simplify the right side.

Tap for more steps...

Step 4.4.2.1

Simplify $\frac{2}{3} (| x - 1 | - \frac{1}{2})$ .

Tap for more steps...

Step 4.4.2.1.1

Apply the distributive property.

$| x + 2 | = \frac{2}{3} | x - 1 | + \frac{2}{3} (- \frac{1}{2})$

Step 4.4.2.1.2

Combine $\frac{2}{3}$ and $| x - 1 |$ .

$| x + 2 | = \frac{2 | x - 1 |}{3} + \frac{2}{3} (- \frac{1}{2})$

Step 4.4.2.1.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.4.2.1.3.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$| x + 2 | = \frac{2 | x - 1 |}{3} + \frac{2}{3} \cdot \frac{- 1}{2}$

Step 4.4.2.1.3.2

Cancel the common factor.

$| x + 2 | = \frac{2 | x - 1 |}{3} + \frac{2}{3} \cdot \frac{- 1}{2}$

Step 4.4.2.1.3.3

Rewrite the expression.

$| x + 2 | = \frac{2 | x - 1 |}{3} + \frac{1}{3} \cdot - 1$

Step 4.4.2.1.4

Combine $\frac{1}{3}$ and $- 1$ .

$| x + 2 | = \frac{2 | x - 1 |}{3} + \frac{- 1}{3}$

Step 4.4.2.1.5

Move the negative in front of the fraction.

$| x + 2 | = \frac{2 | x - 1 |}{3} - \frac{1}{3}$

Step 5

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x + 2 = \pm (\frac{2 | x - 1 |}{3} - \frac{1}{3})$

Step 6

The result consists of both the positive and negative portions of the $\pm$ .

$x + 2 = \frac{2 | x - 1 |}{3} - \frac{1}{3}$

$x + 2 = - (\frac{2 | x - 1 |}{3} - \frac{1}{3})$

Step 7

Solve $x + 2 = \frac{2 | x - 1 |}{3} - \frac{1}{3}$ for $x$ .

Tap for more steps...

Step 7.1

Solve for $| x - 1 |$ .

Tap for more steps...

Step 7.1.1

Rewrite the equation as $\frac{2 | x - 1 |}{3} - \frac{1}{3} = x + 2$ .

$\frac{2 | x - 1 |}{3} - \frac{1}{3} = x + 2$

Step 7.1.2

Move all terms not containing $| x - 1 |$ to the right side of the equation.

Tap for more steps...

Step 7.1.2.1

Add $\frac{1}{3}$ to both sides of the equation.

$\frac{2 | x - 1 |}{3} = x + 2 + \frac{1}{3}$

Step 7.1.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 | x - 1 |}{3} = x + 2 \cdot \frac{3}{3} + \frac{1}{3}$

Step 7.1.2.3

Combine $2$ and $\frac{3}{3}$ .

$\frac{2 | x - 1 |}{3} = x + \frac{2 \cdot 3}{3} + \frac{1}{3}$

Step 7.1.2.4

Combine the numerators over the common denominator.

$\frac{2 | x - 1 |}{3} = x + \frac{2 \cdot 3 + 1}{3}$

Step 7.1.2.5

Simplify the numerator.

Tap for more steps...

Step 7.1.2.5.1

Multiply $2$ by $3$ .

$\frac{2 | x - 1 |}{3} = x + \frac{6 + 1}{3}$

Step 7.1.2.5.2

Add $6$ and $1$ .

$\frac{2 | x - 1 |}{3} = x + \frac{7}{3}$

Step 7.1.3

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 | x - 1 |}{3} = \frac{3}{2} (x + \frac{7}{3})$

Step 7.1.4

Simplify both sides of the equation.

Tap for more steps...

Step 7.1.4.1

Simplify the left side.

Tap for more steps...

Step 7.1.4.1.1

Simplify $\frac{3}{2} \cdot \frac{2 | x - 1 |}{3}$ .

Tap for more steps...

Step 7.1.4.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.1.4.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 | x - 1 |}{3} = \frac{3}{2} (x + \frac{7}{3})$

Step 7.1.4.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 | x - 1 |) = \frac{3}{2} (x + \frac{7}{3})$

Step 7.1.4.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.1.4.1.1.2.1

Factor $2$ out of $2 | x - 1 |$ .

$\frac{1}{2} (2 | x - 1 |) = \frac{3}{2} (x + \frac{7}{3})$

Step 7.1.4.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 | x - 1 |) = \frac{3}{2} (x + \frac{7}{3})$

Step 7.1.4.1.1.2.3

Rewrite the expression.

$| x - 1 | = \frac{3}{2} (x + \frac{7}{3})$

Step 7.1.4.2

Simplify the right side.

Tap for more steps...

Step 7.1.4.2.1

Simplify $\frac{3}{2} (x + \frac{7}{3})$ .

Tap for more steps...

Step 7.1.4.2.1.1

Apply the distributive property.

$| x - 1 | = \frac{3}{2} x + \frac{3}{2} \cdot \frac{7}{3}$

Step 7.1.4.2.1.2

Combine $\frac{3}{2}$ and $x$ .

$| x - 1 | = \frac{3 x}{2} + \frac{3}{2} \cdot \frac{7}{3}$

Step 7.1.4.2.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 7.1.4.2.1.3.1

Cancel the common factor.

$| x - 1 | = \frac{3 x}{2} + \frac{3}{2} \cdot \frac{7}{3}$

Step 7.1.4.2.1.3.2

Rewrite the expression.

$| x - 1 | = \frac{3 x}{2} + \frac{1}{2} \cdot 7$

Step 7.1.4.2.1.4

Combine $\frac{1}{2}$ and $7$ .

$| x - 1 | = \frac{3 x}{2} + \frac{7}{2}$

Step 7.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 1 = \pm (\frac{3 x}{2} + \frac{7}{2})$

Step 7.3

The result consists of both the positive and negative portions of the $\pm$ .

$x - 1 = \frac{3 x}{2} + \frac{7}{2}$

$x - 1 = - (\frac{3 x}{2} + \frac{7}{2})$

Step 7.4

Solve $x - 1 = \frac{3 x}{2} + \frac{7}{2}$ for $x$ .

Tap for more steps...

Step 7.4.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 7.4.1.1

Subtract $\frac{3 x}{2}$ from both sides of the equation.

$x - 1 - \frac{3 x}{2} = \frac{7}{2}$

Step 7.4.1.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \cdot \frac{2}{2} - \frac{3 x}{2} - 1 = \frac{7}{2}$

Step 7.4.1.3

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} - \frac{3 x}{2} - 1 = \frac{7}{2}$

Step 7.4.1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - 3 x}{2} - 1 = \frac{7}{2}$

Step 7.4.1.5

Subtract $3 x$ from $x \cdot 2$ .

Tap for more steps...

Step 7.4.1.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - 3 x}{2} - 1 = \frac{7}{2}$

Step 7.4.1.5.2

Subtract $3 x$ from $2 \cdot x$ .

$\frac{- x}{2} - 1 = \frac{7}{2}$

Step 7.4.1.6

Move the negative in front of the fraction.

$- \frac{x}{2} - 1 = \frac{7}{2}$

Step 7.4.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 7.4.2.1

Add $1$ to both sides of the equation.

$- \frac{x}{2} = \frac{7}{2} + 1$

Step 7.4.2.2

Write $1$ as a fraction with a common denominator.

$- \frac{x}{2} = \frac{7}{2} + \frac{2}{2}$

Step 7.4.2.3

Combine the numerators over the common denominator.

$- \frac{x}{2} = \frac{7 + 2}{2}$

Step 7.4.2.4

Add $7$ and $2$ .

$- \frac{x}{2} = \frac{9}{2}$

Step 7.4.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x = 9$

Step 7.4.4

Divide each term in $- x = 9$ by $- 1$ and simplify.

Tap for more steps...

Step 7.4.4.1

Divide each term in $- x = 9$ by $- 1$ .

$\frac{- x}{- 1} = \frac{9}{- 1}$

Step 7.4.4.2

Simplify the left side.

Tap for more steps...

Step 7.4.4.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{9}{- 1}$

Step 7.4.4.2.2

Divide $x$ by $1$ .

$x = \frac{9}{- 1}$

Step 7.4.4.3

Simplify the right side.

Tap for more steps...

Step 7.4.4.3.1

Divide $9$ by $- 1$ .

$x = - 9$

Step 7.5

Solve $x - 1 = - (\frac{3 x}{2} + \frac{7}{2})$ for $x$ .

Tap for more steps...

Step 7.5.1

Simplify $- (\frac{3 x}{2} + \frac{7}{2})$ .

Tap for more steps...

Step 7.5.1.1

Rewrite.

$x - 1 = 0 + 0 - (\frac{3 x}{2} + \frac{7}{2})$

Step 7.5.1.2

Simplify by adding zeros.

$x - 1 = - (\frac{3 x}{2} + \frac{7}{2})$

Step 7.5.1.3

Apply the distributive property.

$x - 1 = - \frac{3 x}{2} - \frac{7}{2}$

Step 7.5.2

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 7.5.2.1

Add $\frac{3 x}{2}$ to both sides of the equation.

$x - 1 + \frac{3 x}{2} = - \frac{7}{2}$

Step 7.5.2.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \cdot \frac{2}{2} + \frac{3 x}{2} - 1 = - \frac{7}{2}$

Step 7.5.2.3

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} + \frac{3 x}{2} - 1 = - \frac{7}{2}$

Step 7.5.2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 + 3 x}{2} - 1 = - \frac{7}{2}$

Step 7.5.2.5

Add $x \cdot 2$ and $3 x$ .

Tap for more steps...

Step 7.5.2.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x + 3 x}{2} - 1 = - \frac{7}{2}$

Step 7.5.2.5.2

Add $2 \cdot x$ and $3 x$ .

$\frac{5 \cdot x}{2} - 1 = - \frac{7}{2}$

$\frac{5 x}{2} - 1 = - \frac{7}{2}$

Step 7.5.3

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 7.5.3.1

Add $1$ to both sides of the equation.

$\frac{5 x}{2} = - \frac{7}{2} + 1$

Step 7.5.3.2

Write $1$ as a fraction with a common denominator.

$\frac{5 x}{2} = - \frac{7}{2} + \frac{2}{2}$

Step 7.5.3.3

Combine the numerators over the common denominator.

$\frac{5 x}{2} = \frac{- 7 + 2}{2}$

Step 7.5.3.4

Add $- 7$ and $2$ .

$\frac{5 x}{2} = \frac{- 5}{2}$

Step 7.5.3.5

Move the negative in front of the fraction.

$\frac{5 x}{2} = - \frac{5}{2}$

Step 7.5.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$5 x = - 5$

Step 7.5.5

Divide each term in $5 x = - 5$ by $5$ and simplify.

Tap for more steps...

Step 7.5.5.1

Divide each term in $5 x = - 5$ by $5$ .

$\frac{5 x}{5} = \frac{- 5}{5}$

Step 7.5.5.2

Simplify the left side.

Tap for more steps...

Step 7.5.5.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 7.5.5.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 5}{5}$

Step 7.5.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 5}{5}$

Step 7.5.5.3

Simplify the right side.

Tap for more steps...

Step 7.5.5.3.1

Divide $- 5$ by $5$ .

$x = - 1$

Step 7.6

Consolidate the solutions.

$x = - 9, - 1$

Step 8

Solve $x + 2 = - (\frac{2 | x - 1 |}{3} - \frac{1}{3})$ for $x$ .

Tap for more steps...

Step 8.1

Solve for $| x - 1 |$ .

Tap for more steps...

Step 8.1.1

Rewrite the equation as $- (\frac{2 | x - 1 |}{3} - \frac{1}{3}) = x + 2$ .

$- (\frac{2 | x - 1 |}{3} - \frac{1}{3}) = x + 2$

Step 8.1.2

Simplify $- (\frac{2 | x - 1 |}{3} - \frac{1}{3})$ .

Tap for more steps...

Step 8.1.2.1

Apply the distributive property.

$- \frac{2 | x - 1 |}{3} - - \frac{1}{3} = x + 2$

Step 8.1.2.2

Multiply $- - \frac{1}{3}$ .

Tap for more steps...

Step 8.1.2.2.1

Multiply $- 1$ by $- 1$ .

$- \frac{2 | x - 1 |}{3} + 1 (\frac{1}{3}) = x + 2$

Step 8.1.2.2.2

Multiply $\frac{1}{3}$ by $1$ .

$- \frac{2 | x - 1 |}{3} + \frac{1}{3} = x + 2$

Step 8.1.3

Move all terms not containing $| x - 1 |$ to the right side of the equation.

Tap for more steps...

Step 8.1.3.1

Subtract $\frac{1}{3}$ from both sides of the equation.

$- \frac{2 | x - 1 |}{3} = x + 2 - \frac{1}{3}$

Step 8.1.3.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 | x - 1 |}{3} = x + 2 \cdot \frac{3}{3} - \frac{1}{3}$

Step 8.1.3.3

Combine $2$ and $\frac{3}{3}$ .

$- \frac{2 | x - 1 |}{3} = x + \frac{2 \cdot 3}{3} - \frac{1}{3}$

Step 8.1.3.4

Combine the numerators over the common denominator.

$- \frac{2 | x - 1 |}{3} = x + \frac{2 \cdot 3 - 1}{3}$

Step 8.1.3.5

Simplify the numerator.

Tap for more steps...

Step 8.1.3.5.1

Multiply $2$ by $3$ .

$- \frac{2 | x - 1 |}{3} = x + \frac{6 - 1}{3}$

Step 8.1.3.5.2

Subtract $1$ from $6$ .

$- \frac{2 | x - 1 |}{3} = x + \frac{5}{3}$

Step 8.1.4

Multiply both sides of the equation by $- \frac{3}{2}$ .

$- \frac{3}{2} (- \frac{2 | x - 1 |}{3}) = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5

Simplify both sides of the equation.

Tap for more steps...

Step 8.1.5.1

Simplify the left side.

Tap for more steps...

Step 8.1.5.1.1

Simplify $- \frac{3}{2} (- \frac{2 | x - 1 |}{3})$ .

Tap for more steps...

Step 8.1.5.1.1.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.1.5.1.1.1.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$\frac{- 3}{2} (- \frac{2 | x - 1 |}{3}) = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.1.2

Move the leading negative in $- \frac{2 | x - 1 |}{3}$ into the numerator.

$\frac{- 3}{2} \cdot \frac{- 2 | x - 1 |}{3} = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.1.3

Factor $3$ out of $- 3$ .

$\frac{3 (- 1)}{2} \cdot \frac{- 2 | x - 1 |}{3} = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.1.4

Cancel the common factor.

$\frac{3 \cdot - 1}{2} \cdot \frac{- 2 | x - 1 |}{3} = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.1.5

Rewrite the expression.

$\frac{- 1}{2} (- 2 | x - 1 |) = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1.5.1.1.2.1

Factor $2$ out of $- 2 | x - 1 |$ .

$\frac{- 1}{2} (2 (- | x - 1 |)) = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.2.2

Cancel the common factor.

$\frac{- 1}{2} (2 (- | x - 1 |)) = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.2.3

Rewrite the expression.

$- - | x - 1 | = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.3

Multiply.

Tap for more steps...

Step 8.1.5.1.1.3.1

Multiply $- 1$ by $- 1$ .

$1 | x - 1 | = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.1.1.3.2

Multiply $| x - 1 |$ by $1$ .

$| x - 1 | = - \frac{3}{2} (x + \frac{5}{3})$

Step 8.1.5.2

Simplify the right side.

Tap for more steps...

Step 8.1.5.2.1

Simplify $- \frac{3}{2} (x + \frac{5}{3})$ .

Tap for more steps...

Step 8.1.5.2.1.1

Simplify terms.

Tap for more steps...

Step 8.1.5.2.1.1.1

Apply the distributive property.

$| x - 1 | = - \frac{3}{2} x - \frac{3}{2} \cdot \frac{5}{3}$

Step 8.1.5.2.1.1.2

Combine $x$ and $\frac{3}{2}$ .

$| x - 1 | = - \frac{x \cdot 3}{2} - \frac{3}{2} \cdot \frac{5}{3}$

Step 8.1.5.2.1.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.1.5.2.1.1.3.1

Move the leading negative in $- \frac{3}{2}$ into the numerator.

$| x - 1 | = - \frac{x \cdot 3}{2} + \frac{- 3}{2} \cdot \frac{5}{3}$

Step 8.1.5.2.1.1.3.2

Factor $3$ out of $- 3$ .

$| x - 1 | = - \frac{x \cdot 3}{2} + \frac{3 (- 1)}{2} \cdot \frac{5}{3}$

Step 8.1.5.2.1.1.3.3

Cancel the common factor.

$| x - 1 | = - \frac{x \cdot 3}{2} + \frac{3 \cdot - 1}{2} \cdot \frac{5}{3}$

Step 8.1.5.2.1.1.3.4

Rewrite the expression.

$| x - 1 | = - \frac{x \cdot 3}{2} + \frac{- 1}{2} \cdot 5$

Step 8.1.5.2.1.1.4

Combine $\frac{- 1}{2}$ and $5$ .

$| x - 1 | = - \frac{x \cdot 3}{2} + \frac{- 1 \cdot 5}{2}$

Step 8.1.5.2.1.1.5

Multiply $- 1$ by $5$ .

$| x - 1 | = - \frac{x \cdot 3}{2} + \frac{- 5}{2}$

Step 8.1.5.2.1.2

Simplify each term.

Tap for more steps...

Step 8.1.5.2.1.2.1

Move $3$ to the left of $x$ .

$| x - 1 | = - \frac{3 \cdot x}{2} + \frac{- 5}{2}$

Step 8.1.5.2.1.2.2

Move the negative in front of the fraction.

$| x - 1 | = - \frac{3 x}{2} - \frac{5}{2}$

Step 8.2

Remove the absolute value term. This creates a $\pm$ on the right side of the equation because $| x | = \pm x$ .

$x - 1 = \pm (- \frac{3 x}{2} - \frac{5}{2})$

Step 8.3

The result consists of both the positive and negative portions of the $\pm$ .

$x - 1 = - \frac{3 x}{2} - \frac{5}{2}$

$x - 1 = - (- \frac{3 x}{2} - \frac{5}{2})$

Step 8.4

Solve $x - 1 = - \frac{3 x}{2} - \frac{5}{2}$ for $x$ .

Tap for more steps...

Step 8.4.1

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 8.4.1.1

Add $\frac{3 x}{2}$ to both sides of the equation.

$x - 1 + \frac{3 x}{2} = - \frac{5}{2}$

Step 8.4.1.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \cdot \frac{2}{2} + \frac{3 x}{2} - 1 = - \frac{5}{2}$

Step 8.4.1.3

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} + \frac{3 x}{2} - 1 = - \frac{5}{2}$

Step 8.4.1.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 + 3 x}{2} - 1 = - \frac{5}{2}$

Step 8.4.1.5

Add $x \cdot 2$ and $3 x$ .

Tap for more steps...

Step 8.4.1.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x + 3 x}{2} - 1 = - \frac{5}{2}$

Step 8.4.1.5.2

Add $2 \cdot x$ and $3 x$ .

$\frac{5 \cdot x}{2} - 1 = - \frac{5}{2}$

$\frac{5 x}{2} - 1 = - \frac{5}{2}$

Step 8.4.2

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 8.4.2.1

Add $1$ to both sides of the equation.

$\frac{5 x}{2} = - \frac{5}{2} + 1$

Step 8.4.2.2

Write $1$ as a fraction with a common denominator.

$\frac{5 x}{2} = - \frac{5}{2} + \frac{2}{2}$

Step 8.4.2.3

Combine the numerators over the common denominator.

$\frac{5 x}{2} = \frac{- 5 + 2}{2}$

Step 8.4.2.4

Add $- 5$ and $2$ .

$\frac{5 x}{2} = \frac{- 3}{2}$

Step 8.4.2.5

Move the negative in front of the fraction.

$\frac{5 x}{2} = - \frac{3}{2}$

Step 8.4.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$5 x = - 3$

Step 8.4.4

Divide each term in $5 x = - 3$ by $5$ and simplify.

Tap for more steps...

Step 8.4.4.1

Divide each term in $5 x = - 3$ by $5$ .

$\frac{5 x}{5} = \frac{- 3}{5}$

Step 8.4.4.2

Simplify the left side.

Tap for more steps...

Step 8.4.4.2.1

Cancel the common factor of $5$ .

Tap for more steps...

Step 8.4.4.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{- 3}{5}$

Step 8.4.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{5}$

Step 8.4.4.3

Simplify the right side.

Tap for more steps...

Step 8.4.4.3.1

Move the negative in front of the fraction.

$x = - \frac{3}{5}$

Step 8.5

Solve $x - 1 = - (- \frac{3 x}{2} - \frac{5}{2})$ for $x$ .

Tap for more steps...

Step 8.5.1

Simplify $- (- \frac{3 x}{2} - \frac{5}{2})$ .

Tap for more steps...

Step 8.5.1.1

Rewrite.

$x - 1 = 0 + 0 - (- \frac{3 x}{2} - \frac{5}{2})$

Step 8.5.1.2

Simplify by adding zeros.

$x - 1 = - (- \frac{3 x}{2} - \frac{5}{2})$

Step 8.5.1.3

Apply the distributive property.

$x - 1 = - - \frac{3 x}{2} - - \frac{5}{2}$

Step 8.5.1.4

Multiply $- - \frac{3 x}{2}$ .

Tap for more steps...

Step 8.5.1.4.1

Multiply $- 1$ by $- 1$ .

$x - 1 = 1 \frac{3 x}{2} - - \frac{5}{2}$

Step 8.5.1.4.2

Multiply $\frac{3 x}{2}$ by $1$ .

$x - 1 = \frac{3 x}{2} - - \frac{5}{2}$

Step 8.5.1.5

Multiply $- - \frac{5}{2}$ .

Tap for more steps...

Step 8.5.1.5.1

Multiply $- 1$ by $- 1$ .

$x - 1 = \frac{3 x}{2} + 1 (\frac{5}{2})$

Step 8.5.1.5.2

Multiply $\frac{5}{2}$ by $1$ .

$x - 1 = \frac{3 x}{2} + \frac{5}{2}$

Step 8.5.2

Move all terms containing $x$ to the left side of the equation.

Tap for more steps...

Step 8.5.2.1

Subtract $\frac{3 x}{2}$ from both sides of the equation.

$x - 1 - \frac{3 x}{2} = \frac{5}{2}$

Step 8.5.2.2

To write $x$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x \cdot \frac{2}{2} - \frac{3 x}{2} - 1 = \frac{5}{2}$

Step 8.5.2.3

Combine $x$ and $\frac{2}{2}$ .

$\frac{x \cdot 2}{2} - \frac{3 x}{2} - 1 = \frac{5}{2}$

Step 8.5.2.4

Combine the numerators over the common denominator.

$\frac{x \cdot 2 - 3 x}{2} - 1 = \frac{5}{2}$

Step 8.5.2.5

Subtract $3 x$ from $x \cdot 2$ .

Tap for more steps...

Step 8.5.2.5.1

Reorder $x$ and $2$ .

$\frac{2 \cdot x - 3 x}{2} - 1 = \frac{5}{2}$

Step 8.5.2.5.2

Subtract $3 x$ from $2 \cdot x$ .

$\frac{- x}{2} - 1 = \frac{5}{2}$

Step 8.5.2.6

Move the negative in front of the fraction.

$- \frac{x}{2} - 1 = \frac{5}{2}$

Step 8.5.3

Move all terms not containing $x$ to the right side of the equation.

Tap for more steps...

Step 8.5.3.1

Add $1$ to both sides of the equation.

$- \frac{x}{2} = \frac{5}{2} + 1$

Step 8.5.3.2

Write $1$ as a fraction with a common denominator.

$- \frac{x}{2} = \frac{5}{2} + \frac{2}{2}$

Step 8.5.3.3

Combine the numerators over the common denominator.

$- \frac{x}{2} = \frac{5 + 2}{2}$

Step 8.5.3.4

Add $5$ and $2$ .

$- \frac{x}{2} = \frac{7}{2}$

Step 8.5.4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$- x = 7$

Step 8.5.5

Divide each term in $- x = 7$ by $- 1$ and simplify.

Tap for more steps...

Step 8.5.5.1

Divide each term in $- x = 7$ by $- 1$ .

$\frac{- x}{- 1} = \frac{7}{- 1}$

Step 8.5.5.2

Simplify the left side.

Tap for more steps...

Step 8.5.5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{7}{- 1}$

Step 8.5.5.2.2

Divide $x$ by $1$ .

$x = \frac{7}{- 1}$

Step 8.5.5.3

Simplify the right side.

Tap for more steps...

Step 8.5.5.3.1

Divide $7$ by $- 1$ .

$x = - 7$

Step 8.6

Consolidate the solutions.

$x = - \frac{3}{5}, - 7$

Step 9

Consolidate the solutions.

$x = - 9, - 1, - \frac{3}{5}, - 7$

Step 10

Use each root to create test intervals.

$x < - 9$

$- 9 < x < - 7$

$- 7 < x < - 1$

$- 1 < x < - \frac{3}{5}$

$x > - \frac{3}{5}$

Step 11

Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.

Tap for more steps...

Step 11.1

Test a value on the interval $x < - 9$ to see if it makes the inequality true.

Tap for more steps...

Step 11.1.1

Choose a value on the interval $x < - 9$ and see if this value makes the original inequality true.

$x = - 12$

Step 11.1.2

Replace $x$ with $- 12$ in the original inequality.

$3 | (- 12) + 2 | = 2 | (- 12) - 1 | - 1$

Step 11.1.3

The left side $30$ does not equal to the right side $25$ , which means that the given statement is false.

False

Step 11.2

Test a value on the interval $- 9 < x < - 7$ to see if it makes the inequality true.

Tap for more steps...

Step 11.2.1

Choose a value on the interval $- 9 < x < - 7$ and see if this value makes the original inequality true.

$x = - 8$

Step 11.2.2

Replace $x$ with $- 8$ in the original inequality.

$3 | (- 8) + 2 | = 2 | (- 8) - 1 | - 1$

Step 11.2.3

The left side $18$ does not equal to the right side $17$ , which means that the given statement is false.

False

Step 11.3

Test a value on the interval $- 7 < x < - 1$ to see if it makes the inequality true.

Tap for more steps...

Step 11.3.1

Choose a value on the interval $- 7 < x < - 1$ and see if this value makes the original inequality true.

$x = - 4$

Step 11.3.2

Replace $x$ with $- 4$ in the original inequality.

$3 | (- 4) + 2 | = 2 | (- 4) - 1 | - 1$

Step 11.3.3

The left side $6$ does not equal to the right side $9$ , which means that the given statement is false.

False

Step 11.4

Test a value on the interval $- 1 < x < - \frac{3}{5}$ to see if it makes the inequality true.

Tap for more steps...

Step 11.4.1

Choose a value on the interval $- 1 < x < - \frac{3}{5}$ and see if this value makes the original inequality true.

$x = - 0.8$

Step 11.4.2

Replace $x$ with $- 0.8$ in the original inequality.

$3 | (- 0.8) + 2 | = 2 | (- 0.8) - 1 | - 1$

Step 11.4.3

The left side $3.6$ does not equal to the right side $2.6$ , which means that the given statement is false.

False

Step 11.5

Test a value on the interval $x > - \frac{3}{5}$ to see if it makes the inequality true.

Tap for more steps...

Step 11.5.1

Choose a value on the interval $x > - \frac{3}{5}$ and see if this value makes the original inequality true.

$x = 0$

Step 11.5.2

Replace $x$ with $0$ in the original inequality.

$3 | (0) + 2 | = 2 | (0) - 1 | - 1$

Step 11.5.3

The left side $6$ does not equal to the right side $1$ , which means that the given statement is false.

False

Step 11.6

Compare the intervals to determine which ones satisfy the original inequality.

$x < - 9$ False

$- 9 < x < - 7$ False

$- 7 < x < - 1$ False

$- 1 < x < - \frac{3}{5}$ False

$x > - \frac{3}{5}$ False

$x < - 9$ False

$- 9 < x < - 7$ False

$- 7 < x < - 1$ False

$- 1 < x < - \frac{3}{5}$ False

$x > - \frac{3}{5}$ False

Step 12

Since there are no numbers that fall within the interval, this inequality has no solution.

No solution

Step 13

Exclude the solutions that do not make $3 | x + 2 | = 2 | x - 1 | - 1$ true.

$x = - 1, - 7$

Step 14